More number fun

Just remembered another of my maths teacher's little tricks (he was a really good teacher who always kept us interested).  How to prove that 0.9999... (the dots mean the 9s go on for ever) = 1.

x = 0.9999...

10x = 9.9999...

10x - x = 9.9999... - 0.9999....

9x = 9

x = 1

So there you are - or are you?  You may remember the reductio ad absurdum approach which says that if you come up with an absurd or impossible result, then there must be something wrong with your reasoning.  And the tendency is to say that x cannot be both 0.999... and 1 and therefore there must be something wrong.  But in fact this is just a quirk of the way our number system works - irrational numbers (like 0.999...) don't behave the same as "ordinary" numbers and if you do "ordinary" maths with them you are likely to get apparently weird results.